Deterministic Models in Excel Compliments to LargeScale Simulation

Deterministic Models in Excel: Compliments to Large-Scale Simulation CDR Harrison Schramm hcschram@nps. edu 831. 656. 2358 Operations Research Department Naval Postgraduate School, Monterey, CA N 81 Brown Bag 24 July 2012 THIS PRESENTATION IS UNCLASSIFIED

My Intro • N-81 Alumnus, currently on Faculty at NPS • Current work with Deterministic Modeling: – Application to cyber – Applications to Infectious Disease UNCLASSIFIED

Format • • Three Blocks, increasing in technicality Block I : Fundamentals Block II: Next Steps Block III: The Frontiers • After Block I, semi-open ended. UNCLASSIFIED

References: • Aircraft in War: Dawn of the Fourth Arm. F. L. Lanchester • The Pleasures of Counting. T. W. Korner • Epidemic Modeling: Daley and Gani • Lanchester Models of Warfare (vol. 1 and 2), James G. Taylor UNCLASSIFIED

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BLOCK I UNCLASSIFIED

Why are we doing this? • The usefulness of doing “Paper-and-Pencil” analysis – As a supplement to simulations – To guide the right questions! • Fast, Transparent • Where does this not apply? UNCLASSIFIED

Three Steps for mathematical modeling • Tell a story – Draw a picture or use Legos • Write discrete time, discrete space model – How would you play this game with two people and some dice? • Take limits* • Analytic Results* *These steps are not always necessary UNCLASSIFIED

Lanchester Model Story Tabletop / Whiteboard UNCLASSIFIED

Lanchester Models Pit two sides, Blue and Red, against each other, and analyze the resulting combat as a deterministic model. In their most general form, where the gammas represent arbitrary functions. We explore specific choices, and their consequences subsequently UNCLASSIFIED

Common Lanchester Model ‘Flavors’ • For Aimed fire • For Area fire • For Ambush situations UNCLASSIFIED

A note about scaling • Understanding Scaling is important in differential equation models. UNCLASSIFIED

Lanchester Model Vs. Simulation UNCLASSIFIED

Application: Spreadsheet Implementation • We’ll do this in real time. • How it can go wrong – Negative force levels • Extensions and applications: – Reinforcements – Network Application UNCLASSIFIED

Case Study: The battle of Iwo Jima Engel’s Analysis UNCLASSIFIED

Part I Wrap-up • In Block I we discussed: – How to tell a story with mathematics – How to implement this in Microsoft Excel • With added emphasis on: – What can go wrong – Where these methods do not apply UNCLASSIFIED

Block II: Next Steps UNCLASSIFIED

Review: Telling a Story with Math • These are the steps: 1. Tell the story (stick figures, Legos, etc) 2. Write discrete time, discrete space equations 3. See what happens. • In this section, we will tell a new story and look at Lanchester Applications. UNCLASSIFIED

New Model: Infectious Diseases: The S-I and S-I-R Models • The Story: A fixed population of N individuals who interact with each other at some intensity has a pathogen introduced • May be ‘simple’ (S-I) epidemic, or epidemic with removals (S-I-R). UNCLASSIFIED

The Story • Whiteboard UNCLASSIFIED

The Math UNCLASSIFIED

Spreadsheet implementation UNCLASSIFIED

Sapphire Growth as an S-I Process Courtesy: Stefan Savage. DShield is the Distributed Intrusion Detection System Project (www. dshield. org)

Lanchester with Shocks: An application to Networked Forces These slides are shamelessly stolen from my MORS presentation UNCLASSIFIED

Shock Action - modification • Consider a model in which the dynamics of combat change suddenly and irrevocably at a deterministic time, t*. • Our solutions to follow are implicit in the corresponding variables, which we call B* or R* UNCLASSIFIED

The effect of the Network on Targeting • If ordnance errors are equal and uncorrelated, we may say that they are circularly distributed, and Where the common unit of error is Circular Error Probable (The radius that encloses ½ of the rounds fired), which may be converted by: UNCLASSIFIED

Reduction in as a function of CEP UNCLASSIFIED

When should we just switch from Aimed to Area fires? • Let be the firing rate. For Aimed fire: • For Area fire: • We should prefer area fire iff: UNCLASSIFIED

Case Study II: Networked Battle of Iwo Jima UNCLASSIFIED

We may ask… • Suppose that Blue has a vulnerable network, but plans like his network was invulnerable, uses Lanchester for his planning and plans for a 10% casualty rate. • Suppose further that the quality of his network gives him parity with the advantage for being ‘dug in’ • We may ask: What’s the impact of having a his network fail? UNCLASSIFIED

The Impact of Network Failure UNCLASSIFIED

Part II Wrap-up • In this section, we: – Derived the model for infectious diseases from first principles – Applied in a spreadsheet – Showed how Lanchester models may be adapted for Cyber Effects. UNCLASSIFIED

Block III: The Frontiers This section contains current research. UNCLASSIFIED

S-I and Stuxnet UNCLASSIFIED

Applying S-I model to Stuxnet… Unclassified data from W. 32 Stuxnet Dossier, Symantec Corporation White Paper Stuxnet Propagation by Country 40000 35000 Machines Infected 30000 Iran indonesia 25000 India Azerbaijan 20000 Pakistan Malaysia 15000 USA Uszbekistan 10000 Russia Great Britain 5000 0 0. 00 50. 00 100. 00 150. 00 200. 00 250. 00 300. 00 Days since zero 350. 00 400. 00 450. 00 500. 00

Best-fit Cross-Infectivity Rates This is a notional sketch to show what you could do with this data if you had it. UNCLASSIFIED

Stochastic Lanchester UNCLASSIFIED

Lanchester Equations: A probabilistic Approach • We said earlier that we’re using the Expected value (or mean field) approximation to the process. • Expectation of what? • Following the assumptions of the Lanchester Model, the Distribution for the blue losses in a ‘small’ interval is UNCLASSIFIED

Stochastic Diffusions and Lanchester • We may consider this as a stochastic diffusion, with the Stochastic Differential Equations: • Which lead to the Ordinary Differential Equations: UNCLASSIFIED

Variance Dashed lines are simulation, Solid lines SDEs UNCLASSIFIED

Covariance UNCLASSIFIED

Block III Wrap up • In this section, we moved ‘into the frontiers’: – Cyber Applications of S-I – Stochastic Lanchester • Thank you for your time and interest. UNCLASSIFIED

Fin. UNCLASSIFIED

Backups UNCLASSIFIED

Aimed Fire → Aimed Fire Model and results • In this situation network loss causes us to go from highly effective aimed fire to less accurate aimed fire. The model is specified as: UNCLASSIFIED

Aimed Fire → Area Fire: Model and Result • Conversely, in this situation, network reduction causes us to go from aimed fire to area fire UNCLASSIFIED

You could also do this… Stuxnet infectivity parameters (Least Squares 0. 018 Fit) 0. 016 0. 014 0. 012 Iran 0. 01 indonesia India 0. 008 0. 006 0. 004 0. 002 n Ira es on in d In di a ia n Az er ba ija n ta kis Pa sia ay al M A US zb e n kist Us ss ia Ru G Br rea ita t in a 0 Azerbaijan Pakistan Malaysia USA Uszbekistan Russia Great Britain